Exact and approximate nonlinear waves generated by the periodic superposition of solitons

Toda [1], Boyd [2], Zaitsev [3], Korpel & Banerjee [4], and Whitham [5] have proved that many species of solitons may be cloned and superposed with even spacing to generateexact nonlinear, spatially periodic solutions ("cnoidal waves"). The equations solved by such "imbricate" series of solitary waves include the Korteweg-deVries, Cubic Schroedinger, Benjamin-Ono, and resonant triad equations. However, all existing theorems apply only when the solitons arerational ormeromorphic functions and the cnoidal waves areelliptic functions. In this note, we ask: does the exact soliton-superposition apply to non-elliptic solitons and cnoidal waves? Although a complete answer to this (very broad!) question eludes us, it is possible to offer a revealing counterexample. The quartic Korteweg-deVries equation has solutions which arehyperelliptic, and thus very special. Nevertheless, its periodic solutions are not the exact superposition of the infinite number of copies of a soliton. This is highly suggestive that non-elliptic extensions of the Toda theorem are rare or non-existent. It is intriguing, however, that the soliton-superposition generates a very goodapproximation to the hypercnoidal wave even when the solitons strongly overlap.